Adaptive Control of Parabolic Pdes

by Smyshlyaev, Andrey; Krstic, Miroslav

• ISBN13: 9781400835362
• ISBN10: 1400835364
• Copyright: 2010-06-03
• Publisher: Princeton Univ Pr

Summary

This book introduces a comprehensive methodology for adaptive control design of parabolic partial differential equations with unknown functional parameters, including reaction-convection-diffusion systems ubiquitous in chemical, thermal, biomedical, aerospace, and energy systems. Andrey Smyshlyaev and Miroslav Krstic develop explicit feedback laws that do not require real-time solution of Riccati or other algebraic operator-valued equations. The book emphasizes stabilization by boundary control and using boundary sensing for unstable PDE systems with an infinite relative degree. The book also presents a rich collection of methods for system identification of PDEs, methods that employ Lyapunov, passivity, observer-based, swapping-based, gradient, and least-squares tools and parameterizations, among others.Including a wealth of stimulating ideas and providing the mathematical and control-systems background needed to follow the designs and proofs, the book will be of great use to students and researchers in mathematics, engineering, and physics. It also makes a valuable supplemental text for graduate courses on distributed parameter systems and adaptive control.

Table of Contents

Preface	p. ix
Introduction	p. 1
Parabolic and Hyperbolic PDE Systems	p. 1
The Roles of PDE Plant Instability, Actuator Location, Uncertainty Structure, Relative Degree, and Functional Parameters	p. 2
Class of Parabolic PDE Systems	p. 3
Backstepping	p. 4
Explicitly Parametrized Controllers	p. 5
Adaptive Control	p. 5
Overview of the Literature on Adaptive Control for Parabolic PDEs	p. 6
Inverse Optimality	p. 7
Organization of the Book	p. 7
Notation	p. 9
Nonadaptive Controllers	p. 11
State Feedback	p. 13
Problem Formulation	p. 13
Backstepping Transformation and PDE for Its Kernel	p. 14
Converting the PDE into an Integral Equation	p. 17
Analysis of the Integral Equation by Successive Approximation Series	p. 19
Stability of the Closed-Loop System	p. 22
Dirichlet Uncontrolled End	p. 24
Neumann Actuation	p. 26
Simulation	p. 27
Discussion	p. 27
Notes and References	p. 33
Closed-Form Controllers	p. 35
The Reaction-Diffusion Equation	p. 35
A Family of Plants with Spatially Varying Reactivity	p. 38
Solid Propellant Rocket Model	p. 40
Plants with Spatially Varying Diffusivity	p. 42
The Time-Varying Reaction Equation	p. 45
More Complex Systems	p. 50
2D and 3D Systems	p. 52
Notes and References	p. 54
Observers	p. 55
Observer Design for the Anti-Collocated Setup	p. 55
Plants with Dirichlet Uncontrolled End and Neumann Measurements	p. 58
Observer Design for the Collocated Setup	p. 59
Notes and References	p. 61
Output Feedback	p. 63
Anti-Collocated Setup	p. 63
Collocated Setup	p. 65
Closed-Form Compensators	p. 67
Frequency Domain Compensator	p. 71
Notes and References	p. 72
Control of Complex-Valued PDEs	p. 73
State-Feedback Design for the Schrödinger Equation	p. 73
Observer Design for the Schrödinger Equation	p. 76
Output-Feedback Compensator for the Schrödinger Equation	p. 79
The Ginzburg-Landau Equation	p. 81
State Feedback for the Ginzburg-Landau Equation	p. 83
Observer Design for the Ginzburg-Landau Equation	p. 98
Output Feedback for the Ginzburg-Landau Equation	p. 101
Simulations with the Nonlinear Ginzburg-Landau Equation	p. 104
Notes and References	p. 107
Adaptive Schemes	p. 109
Systematization of Approaches to Adaptive Boundary Stabilization of PDEs	p. 111
Categorization of Adaptive Controllers and Identifiers	p. 111
Benchmark Systems	p. 113
Controllers	p. 114
Lyapunov Design	p. 115
Certainty Equivalence Designs	p. 117
Trade-offs between the Designs	p. 121
Stability	p. 122
Notes and References	p. 124
Lyapunov-Based Designs	p. 125
Plant with Unknown Reaction Coefficient	p. 125
Proof of Theorem 8.1	p. 128
Well-Posedness of the Closed-Loop System	p. 132
Parametric Robustness	p. 134
An Alternative Approach	p. 135
Other Benchmark Problems	p. 136
Systems with Unknown Diffusion and Advection Coefficients	p. 142
Simulation Results	p. 147
Notes and References	p. 149
Certainty Equivalence Design with Passive Identifiers	p. 150
Benchmark Plant	p. 150
3D Reaction-Advection-Diffusion Plant	p. 154
Proof of Theorem 9.2	p. 157
Simulations	p. 163
Notes and References	p. 164
Certainty Equivalence Design with Swapping Identifiers	p. 166
Reaction-Advection-Diffusion Plant	p. 166
Proof of Theorem 10.1	p. 169
Simulations	p. 175
Notes and References	p. 175
State Feedback for PDEs with Spatially Varying Coefficients	p. 176
Problem Statement	p. 176
Nominal Control Design	p. 177
Robustness to Error in Gain Kernel	p. 179
Lyapunov Design	p. 185
Lyapunov Design for Plants with Unknown Advection and Diffusion Parameters	p. 190
Passivity-Based Design	p. 191
Simulations	p. 195
Notes and References	p. 197
Closed-Form Adaptive Output-Feedback Contollers	p. 198
Lyapunov Design--Plant with Unknown Parameter in the Domain	p. 199
Lyapunov Design--Plant with Unknown Parameter in the Boundary Condition	p. 205
Swapping Design--Plant with Unknown Parameter in the Domain	p. 210
Swapping Design--Plant with Unknown Parameter in the Boundary Condition	p. 216
Simulations	p. 223
Notes and References	p. 225
Output Feedback for PDEs with Spatially Varying Coefficients	p. 226
Reaction-Advection-Diffusion Plant	p. 226
Transformation to Observer Canonical Form	p. 227
Nominal Controller	p. 228
Filters	p. 230
Frequency Domain Compensator with Frozen Parameters	p. 232
Update Laws	p. 233
Stability	p. 235
Trajectory Tracking	p. 242
The Ginzburg-Landau Equation	p. 244
Identifier for the Ginzburg-Landau Equation	p. 246
Stability of Adaptive Scheme for the Ginzburg-Landau Equation	p. 248
Simulations	p. 255
Notes and References	p. 255
Inverse Optimal Control	p. 261
Nonadaptive Inverse Optimal Control	p. 262
Reducing Control Effort through Adaptation	p. 265
Dirichlet Actuation	p. 267
Design Example	p. 267
Comparison with the LQR Approach	p. 268
Inverse Optimal Adaptive Control	p. 271
Stability and Inverse Optimality of the Adaptive Scheme	p. 273
Notes and References	p. 275
Adaptive Backstepping for Nonlinear ODEs--The Basics	p. 277
Nonadaptive Backstepping--The Known Parameter Case	p. 277
Tuning Functions Design	p. 279
Modular Design	p. 289
Output Feedback Designs	p. 297
Extensions	p. 303
Poincaré and Agmon Inequalities	p. 305
Bessel Functions	p. 307
Bessel Function Jn	p. 307
Modified Bessel Function In	p. 307
Barbalat's and Other Lemmas for Proving Adaptive Regulation	p. 310
Basic Parabolic PDEs and Their Exact Solutions	p. 313
Reaction-Diffusion Equation with Dirichlet Boundary Conditions	p. 313
Reaction-Diffusion Equation with Neumann Boundary Conditions	p. 315
Reaction-Diffusion Equation with Mixed Boundary Conditions	p. 315
References	p. 317
Index	p. 327
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